PRCTwoAdicAxisTwistRatioCharacter_of_two_three_local_orientation_failure_character
plain-language theorem explainer
A 2·3 composite-local orientation failure witness yields an uncalibrated two-adic axis-twist ratio character. Native-cost uniqueness and prime-calibration arguments cite this bridge when reducing failure surfaces to the thinner axis-twist target. The proof is a one-line existential projection that drops the local-orientation negation.
Claim. If there exists a map $\chi$ on ratio orbits that is a ratio character, twists the two-adic axis, and fails the $2\cdot 3$ composite-local orientation condition, then there exists a ratio character that twists the two-adic axis.
background
In the Primitive Recognition Calculus native-cost uniqueness development, ratio characters are maps on ratio orbits that encode admissible branch behavior for the cost functional. The two-adic axis twist is the branch pattern that sends the orbit of $2$ to the reciprocal branch while keeping other structure under control.
The failure character is the constructive countermodel surface for the $2\cdot 3$ composite-local orientation condition: a ratio character that already carries the two-adic axis twist and additionally violates composite-local orientation at $2\cdot 3$. The reduced target keeps only the ratio-character and axis-twist conjuncts. As the failure-character doc states, that surface is "equivalent to the reduced two-adic ratio-character target."
The axis-twist target is uncalibrated: Pass 115 later shows prime-calibration fields become automatic once a ratio character carries this branch behavior.
proof idea
Term-mode projection. Unpack the failure witness as an existential package $(\chi, h_\chi, h_{\mathrm{branch}}, _)$ consisting of a ratio character, the two-adic axis-twist property, and a discarded local-orientation failure conjunct. Repackage $(\chi, h_\chi, h_{\mathrm{branch}})$ as the thinner axis-twist ratio-character target. No further lemmas are applied.
why it matters
This is the standard reduction step from the $2\cdot 3$ composite-local failure surface to the uncalibrated two-adic axis-twist ratio character. Downstream, it feeds the prime-calibrated axis-twist lift, several "not of failure character" negations of prime-calibration consistency targets (prime-identity forcing, prime-pair product cost consistency, two-prime mixed composite cost consistency), the two-three composite local fork certificate, the iff linking failure to axis-twist ratio character, and absurdity results under prime-identity branch uniformity or the local-orientation target.
In the Recognition framework it sits inside native-cost uniqueness for the PRC cost, the layer that forces the J-cost shape (T5) and the self-similar fixed point $\phi$ (T6) once branch characters are pinned. It does not itself force uniqueness; it thins the countermodel interface so later calibration and fork arguments can fire cleanly.
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